Banach–Tarski paradox

The Banach - Tarski paradox or even Banach and Tarski is a statement of mathematics, which demonstrates that the descriptive term volume can not be generalized to arbitrary sets of points. Then you can break down a ball in three or more dimensions such that its parts can be put together to complete two loose balls again, each of which has the same diameter as the original. The volume doubled, without being clearly seen that, as can arise from this procedure volume from nothing should. This paradox demonstrates that the mathematical model of space as a set of points in mathematics has aspects that are not reflected in physical reality.

Explanation

The paradox is explained mathematically formally so that the ball parts are so complicated shaped such that its volume is no longer in a suitable sense can be defined. More precisely: It is impossible to define on the set of all subsets of the three-dimensional space a bewegungsinvarianten content, the balls a volume equal to zero or infinity assigns. A table of contents is a function that called each quantity of a predetermined range of amounts of a positive real number or infinity assigns volume of the crowd, in particular so that the volume of the union of two non-intersecting volumes equal to the sum of the volumes of the individual quantities. A table of contents is bewegungsinvariant when the volume of a quantity under rotations, shifts and reflections not changed. Each volume mathematical term that should be a bewegungsinvarianter content or even a bewegungsinvariantes measure, therefore, must be restricted so that it is not defined for certain quantities, such as this quantities, in which the ball can be disassembled. Approximately one defines the volume then only for amounts that are the Borel σ - algebra or are Lebesgue measurable. This does not include these amounts. They are infinitely delicate and porous or dust- cloud-like in a sense. The mathematical existence of such amounts is not self-evident: To prove the existence of non-measurable subsets in the -dimensional real space one needs the axiom of choice or weaker forms thereof, which are not derivable from the Zermelo -Fraenkel set theory. Measurable sets of points, however, are additive in terms of volume.

The Polish mathematician Stefan Banach and Alfred Tarski in 1924 conducted a mathematical proof of existence and showed that in the case of the ball a decomposition into six parts was sufficient. Impossible, however, is a constructive proof in the sense of an instruction, such as a ball is actually cut into six pieces to assemble these same two balls in volume can.

General formulation

In a more general formulation of this theorem, the source and end body can differ by an arbitrary volume factor and also have any different shape to certain limitations. The general formulation of this mathematical set in rooms with three or more dimensions is:

Be an integer and be bounded sets with non- empty interior. Then there is a natural number and a disjoint decomposition of associated movements and such that is the disjoint union of the sets.

Situation in one and two dimensions

In the plane and on the line of this sentence is not valid. There are bewegungsinvariante content on the set of all subsets that map circles or lines their usual surface areas or lengths. However, these play only a minor role in mathematics because they are not clearly defined firstly by the surface areas of circles and lengths of lines, and they do not contain dimensions, that is, the union of countably many sets that do not overlap, has sometimes a different content than the sum (in the sense of a series ) of the individual contents. This property of moderation but it is required in very many situations, which is why you generally satisfied also in the one and two dimensional with content that are defined only on certain subsets, but even have measurements. The non-existence of a bewegungsinvarianten measure on the set of all subsets of the line or plane is shown (using the axiom of choice ) by the set of Vitali with the existence of so-called Vitali sets.

1990 Miklós Laczkovich was able to show that at least one similar to the above sentence applies for some areas, but without the " paradox " of a change in volume. Then two equal areas with sufficiently smooth boundary are also equidecomposable. In this sense, a quadrature of the circle is possible, for example, though not with ruler and compass. The number of parts required for a constructive solution was estimated by Laczkovich on about, where the sizes of the larger fragments were not clearly defined by Laczkovich.

However, without some form of the axiom of choice, the theorem can not be proven. Robert M. Solovay could show in 1970 under the assumption of the existence of an inaccessible cardinal number that a model of Zermelo -Fraenkel set theory exists in which all quantities are Lebesgue measurable. It is even possible to maintain the validity of a weaker version of the axiom of choice, namely the axiom DC, which is sufficient for many proofs of elementary calculus. In addition, could be achieved in this model is that every subset of the real numbers has the Baire property and that any uncountable subset of the real numbers contains a nonempty perfect subset. These two statements contradict the general axiom of choice.